Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions

1977 ◽  
Vol 29 (5) ◽  
pp. 937-946
Author(s):  
Hock Ong
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Transformation ◽  
Matrix Function ◽  
Multiplicative Group ◽  
Matrix Functions ◽  
Linear Transformations ◽  
General Matrix ◽  
The Matrix ◽  
Image Position

Let F be a field, F* be its multiplicative group and Mn(F) be the vector space of all n-square matrices over F. Let Sn be the symmetric group acting on the set {1, 2, … , n}. If G is a subgroup of Sn and λ is a function on G with values in F, then the matrix function associated with G and X, denoted by Gλ, is defined byand letℐ(G, λ) = { T : T is a linear transformation of Mn(F) to itself and Gλ(T(X)) = Gλ(X) for all X}.

Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions

1967 ◽  
Vol 19 ◽  
pp. 281-290 ◽  
Author(s):  
E. P. Botta
Keyword(s):  
Vector Space ◽  
Symmetric Function ◽  
Elementary Symmetric Function ◽  
Matrix Functions ◽  
Linear Transformations ◽  
General Matrix ◽  
Linear Maps ◽  
Generalized Matrix ◽  
Image Position

Let Mm(F) be the vector space of m-square matriceswhere F is a field; let f be a function on Mm(F) to some set R. It is of interest to determine the linear maps T: Mm(F) → Mm(F) which preserve the values of the function ƒ; i.e., ƒ(T(X)) = ƒ(X) for all X. For example, if we take ƒ(X) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank. Other classical invariants that may be taken for f are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues. Dieudonné (1), Hua (2), Jacobs (3), Marcus (4, 6, 8), Mori ta (9), and Moyls (6) have conducted extensive research in this area. A class of matrix functions that have recently aroused considerable interest (4; 7) is the generalized matrix functions in the sense of I. Schur (10).

Linear Transformation on Matrices: The Invariance of a Class of General Matrix Functions. II

1968 ◽  
Vol 20 ◽  
pp. 739-748 ◽  
Author(s):  
Peter Botta
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Symmetric Function ◽  
Elementary Symmetric Function ◽  
Matrix Functions ◽  
General Matrix ◽  
Linear Maps

Let Mm(F) be the vector space of m-square matrices X — (Xij), i,j= 1, … , m over a field ƒ;ƒ a function on Mm(F) to some set R. It is of interest to determine the structure of the linear maps T: Mm(F) → Mm(F) that preserve the values of the function ƒ (i.e., ƒ(T(x)) — ƒ(x) for all X). For example, if we take ƒ(x) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank (6). Other classical invariants that may be taken for ƒ are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues.

scholarly journals Rank k Vectors in Symmetry Classes of Tensors

1976 ◽  
Vol 19 (1) ◽  
pp. 67-76 ◽  
Author(s):  
Ming-Huat Lim
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Multiplicative Group ◽  
Vector Spaces ◽  
Linear Character ◽  
Symmetry Classes ◽  
Multilinear Function ◽  
Image Position

Let F be a field, G a subgroup of Sm, the symmetric group of degree m, and χ a linear character on G, i.e., a homomorphism of G into the multiplicative group of F. Let V1,...,Vm be vector spaces over F such that Vi = Vσ(i) for i=1,…,m and for all σ∈G. If W is a vector space over F, then a m-multilinear function is said to be symmetric with respect to G and χ iffor any σ ∊ G and for arbitrary xi ∊ Vi.

Generalization of Gracia's Results

2015 ◽  
Vol 30 ◽  
Author(s):  
Jun Liao ◽  
Heguo Liu ◽  
Yulei Wang ◽  
Zuohui Wu ◽  
Xingzhong Xu
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Matrix Equations ◽  
Multilinear Algebra ◽  
Dimensional Vector ◽  
Complex Field ◽  
Linear Transformations ◽  
Complex Matrices ◽  
Dimensional Vector Space ◽  
The Matrix

Let α be a linear transformation of the m × n-dimensional vector space M_{m×n}(C) over the complex field C such that α(X) = AX −XB, where A and B are m×m and n×n complex matrices, respectively. In this paper, the dimension formulas for the kernels of the linear transformations α^2 and α^3 are given, which generalizes the work of Gracia in [J.M. Gracia. Dimension of the solution spaces of the matrix equations [A, [A, X]] = 0 and [A[A, [A, X]]] = 0. Linear and Multilinear Algebra, 9:195–200, 1980.].

Mean Value and Limit Theorems for Generalized Matrix Functions

1969 ◽  
Vol 21 ◽  
pp. 982-991 ◽  
Author(s):  
Paul J. Nikolai
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Limit Theorems ◽  
Mean Value ◽  
Complex Numbers ◽  
Matrix Functions ◽  
Principal Submatrix ◽  
Generalized Matrix ◽  
Image Position ◽  
Square Matrix

Let A = [aij] denote an n-square matrix with entries in the field of complex numbers. Denote by H a subgroup of Sn, the symmetric group on the integers 1, …, n, and by a character of degree 1 on H. Thenis the generalized matrix function of A associated with H and x; e.g., if H = Sn and χ = 1, then the permanent function. If the sequences ω = (ω1, …, ωm) and ϒ = (ϒ1, …, ϒm) are m-selections, m ≦ w, of integers 1, …, n, then A [ω| ϒ] denotes the m-square generalized submatrix [aωiϒj], i, j = 1, …, m, of the n-square matrix A. If ω is an increasing m-combination, then A [ω|ω] is an m-square principal submatrix of A.

scholarly journals An Algorithm for the Permanent of Circulant Matrices

1977 ◽  
Vol 20 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Larry J. Cummings ◽  
Jennifer Seberry Wallis
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Permutation Matrix ◽  
Circulant Matrices ◽  
The Matrix ◽  
Image Position

The permanent of an n ✕ n matrix A = (aij) is the matrix function1where the summation is over all permutations in the symmetric group, Sn. An n ✕ n matrix A is a circulant if there are scalars a1 …, an such that2where P is the n ✕ n permutation matrix corresponding to the cycle (12 … n) in Sn.

Nonzero Symmetry Classes of Smallest Dimension

1980 ◽  
Vol 32 (4) ◽  
pp. 957-968 ◽  
Author(s):  
G. H. Chan ◽  
M. H. Lim
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Mapping ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Complex Character ◽  
Tensor Power ◽  
Dimensional Vector Space ◽  
Symmetry Classes ◽  
Image Position

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such thatfor all x1, …, xm in U.Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizeris a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

scholarly journals A Norm Inequality for Linear Transformations

1961 ◽  
Vol 4 (3) ◽  
pp. 239-242
Author(s):  
B.N. Moyls ◽  
N.A. Khan
Keyword(s):  
Positive Integer ◽  
Vector Space ◽  
Hermitian Operator ◽  
Norm Inequality ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Image Position ◽  
Ky Fan

In 1949 Ky Fan [1] proved the following result: Let λ1…λn be the eigenvalues of an Hermitian operator H on an n-dimensional vector space Vn. If x1, …, xq is an orthonormal set in V1, and q is a positive integer such n that 1 ≤ q ≤ n, then1

Solvable and Nilpotent Subgroups of GL(n,qm)

1982 ◽  
Vol 34 (5) ◽  
pp. 1097-1111 ◽  
Author(s):  
Thomas R. Wolf
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Solvable Subgroup ◽  
Linear Transformations ◽  
Completely Reducible ◽  
Image Position

Let V ≠ 0 be a vector space of dimension n over a finite field of order qm for a prime q. Of course, GL(n, qm) denotes the group of -linear transformations of V. With few exceptions, GL(n, qm) is non-solvable. How large can a solvable subgroup of GL(n, qm) be? The order of a Sylow-q-subgroup Q of GL(n, qm) is easily computed. But Q cannot act irreducibly nor completely reducibly on V.Suppose that G is a solvable, completely reducible subgroup of GL(n, qm). Huppert ([9], Satz 13, Satz 14) bounds the order of a Sylow-q-subgroup of G, and Dixon ([5], Corollary 1) improves Huppert's bound. Here, we show that |G| ≦ q3nm = |V|3. In fact, we show thatwhere

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